1. Field of the Invention
The present invention relates to a measuring apparatus, optical system manufacturing method, exposure apparatus, device manufacturing method, and processing apparatus.
2. Description of the Related Art
In image processing or signal processing, sampled data (sampling data) is sometimes represented using orthogonal functions (orthogonal polynomials). For example, in the field of optics, the wavefront aberration of the projection optical system of an exposure apparatus is represented using Zernike polynomials as orthogonal functions of pupil coordinates.
There are roughly two reasons why fitting of Zernike polynomials is used in the field of optics. The first is that the Zernike polynomials are orthogonal in a circular region. The second is that the terms of Zernike polynomials correspond to the aberrations of an optical system, and easy handling based on the aberration theory is possible. The terms (y-θ coordinates) of the Zernike polynomials are represented byZ1=1Z2=r cos(θ)Z3=r sin(θ)Z4=2r2−1Z5=r2 cos(2θ)Z6=r2 sin(2θ)Z7=(3r3−2r)cos(θ)Z8=(3r3−2r)sin(θ)Z9=(6r4−6r2+1)Z10=r3 cos(3θ)Z11=r3 sin(3θ)Z12=(4r4−3r2)cos(2θ)Z13=(4r4−3r2)sin(2θ)Z14=(10r5−12r3+3r)cos(θ)Z15=(10r5−12r3+3r)sin(θ)Z16=(20r6−30r4+12r2−1)Z17=r4 cos(4θ)Z18=r4 sin(4θ)Z19=(5r5−4r3)cos(3θ)Z20=(5r5−4r3)sin(3θ)Z21=(15r6−20r4+6r2)cos(2θ)Z22=(15r6−20r4+6r2)sin(2θ)Z23=(35r7−60r5+30r3−4r)cos(0)Z24=(35r7−60r5+30r3−4r)sin(θ)Z25=(70r8−140r6+90r4−20r2+1)Z26=r5 cos(5θ)Z27=r5 sin(5θ)Z28=(6r6−5r4)cos(4θ)Z29=(6r6−5r4)sin(4θ)Z30=(21r7−30r5+10r3)cos(3θ)Z31=(21r7−30r5+10r3)sin(3θ)Z32=(56r8−105r6+60r4−10r2)cos(2θ)Z33=(56r8−105r6+60r4−10r2)sin(20)Z34=(126r9−280r7+210r5−60r3+5r)cos(θ)Z35=(126r9−280r7+210r5−60r3+5r)sin(θ)Z36=(252r10−630r8+560r6−210r4+30r2−1)
The wavefront aberration of the projection optical system is generally measured using an interferometer, as disclosed in Japanese Patent Laid-Open Nos. 2000-277411, 2002-022608, and 2008-277632. For example, when a Fizeau interferometer is used, an image sensor having an imaging plane formed from a plurality of pixels senses interference fringes between detection target light transmitted through the projection optical system and reference light reflected by the Fizeau surface. The phase is calculated from the strength of the interference fringes sensed by the image sensor, and Zernike polynomials are fitted to obtain the wavefront aberration of the projection optical system.
In the conventional technique, accurate phase calculation may be impossible because of, for example, degradation in the visibility of interference fringes caused by disturbance or defects of the interferometer itself (for example, manufacturing errors in the optical system or stage errors). In this case, pixels that have failed in accurate phase calculation are made invalid (missing points). Abnormality in the image data (sampling data) of each pixel is detected based on the number of valid pixels (that is pixels that have succeeded in accurate phase calculation).
However, in abnormality detection based on the number of valid pixels, for example, the reliability (orthogonality) of sampling data for the Zernike polynomials serving as evaluation functions cannot be evaluated. More specifically, even if the number of missing points does not change, the generated error amount changes because a missing point in the periphery of the pupil and a missing point at the center of the pupil have different contributions to the Zernike polynomials. That is, abnormality detection by the number of valid pixels lacks strictness. In addition, when fitting orthogonal functions to sampling data containing missing points, it is necessary to evaluate the reliability of the sampling data for the orthogonal functions, not only concerning the wavefront aberration of the projection optical system.